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NEW YORK UNIVERSITY

INSTITUTE CF MATHEMATICAL SCIENCES

25 Wiverly Place, New York 3, N. "Â«.

NEW YORK UNIVERSITY

INSTITUTE OF MATHEMATICAL SCIENCES

DIVISION OF ELECTROMAGNETIC RESEARCH I

RESEARCH REPORT No. EM-58 !

MULTIPLE SCATTERING OF WAVES

BY PLANAR RANDOM DISTRIBUTIONS

OF PARALLEL CYLINDERS AND BOSSES

by

VIC TWERSKY

CONTRACT No. AF-19(122)-42

OCTOBER 1953

NEW YORK UNIVERSITY

Institute of Mathematical Sciences

Division of Electromagnetic Research

Research Report No. EM-58

MULTIPLE SCATTERING OF WAVES BY PLANAR RANDOM DISTRIBUTIONS

OF PARALLEL CYLINDERS AND BOSSES

by

Vic Twersky

Vi,c 'I'wersky^

Morris Kiine

Project Director

The research reported in this document has been made possible through

support and sponsorship extended by the Air Force Cambridge Research

Center, under Contract No. AF-19(122)-42. It is published for technical

information only, and does not necessarily represent recommendations

or conclusions of the sponsoring agency.

October, 1953

New York, 1953

Manufactured in the United States for New York University Press

by the University's Office of Publications and Printing

IA3LE or CONTEIITS

P8^

Abstract

1, Introduction ^

2, Preliminary Consideration and Statement of the

Problem ^

2.1 Scattering "by a Planar Configuration of

Parallel Cylinders 3

2.2 The Ensemljle of Configurations 9

2.3 Statement of the Problem ^3

3, The Average Wave Function ^^

3.1 Derivation of a Closed- form Approximation

for the Scattering Coefficients 17

3.2 Analysis of the Functions Involved 21

3.3 The Range of Validity of the Closed-fona

Representation 28

U, The Average Intensity 32

5. The Average Energy Flux ^1

6, Distribution of Bosses on a Perfectly Reflecting

Plane

Appendix

Abstract

56

The two-dimensional problem of multiple scattering of a plane

wave by a planar 'random' distribution of parallel cylinders is considered.

A formal solution,> 0, then the roots siaj' he approxi-

Tuated hy v^i^ (2WTi + c)/h. We find e = -i2inTp6, w^ ~ i2wrrp, and

consequently (l^) reduces to

00 iu2npu â–

(16) p(u)^p+^ ^^^iuznph â€¢ u>h^h.

tf= +1

We refer to (l6) as the 'more ordered' case. In the limit 6 â€” > 0, (l6)

reduces to the results for a diffraction grating of spacing "b.

2.3 Stfttftment of the Frohlem

We are now in a position to state explicitly the prohlem

considered in this paper. We are given the wave scattered hy a

single configuration of cylinders, i.e., xy = ^>/-Â» where vj/^ as

given "by (3) and (U) is expressed in terms of the known solution

for the single cylinder. We seek to average \|r and |\|;| over the

ensemhle specified "by the distrilmtion f-anction

*2^^8Â» ^s'^ ^ ^l^'^s^ ^^^s'^'s'^ ^Â°^ pairs where W^ and P are ^iven hy ^10)

and (ll). We also seek closed-form approximations for the averages

which may "be applied to the physical prohlem of scattered reflection

from a striated surface.

Sections 3 ^^^ ^ are concerned with the average values

of >p and |\|f| respectively. Section 5 deals with the average

energy fliix and certain energy theorems. Section 6 treats the

analogous proTslem for "bosses on a perfectly reflecting plane.

In the following sections we use either a "bar or angular

"brackets to indicate an average value.

^Ik-

3Â» The Average Wave S anction

The average value of the scattered wave defined "by (l),

(3) axid (^) niay he written as

8 3 J J

(17)

= E Z ^"^ / \ ^^8 ^^^ (-i'>Q^3sina)H^(fcr^ )e3p (inO^ ) < B^^>^ dy^

an "^ â€” d

where < B > indicates that B as in (^) is to he averaged over all

1^ s ns ^^

variahles except y , iÂ»e., over all configurations assumed hy cylinders

s' ^ 8. We refer to < > as an average with one variable held fixed.

(We use essentially the same notation and nomenclature as Foldy and

n

"Lblx â€¢ ) Similarly we have

^^, dy^. 1

where < B â€¢> t is the average scattering coefficient with two variables

OS SS '

held fixed.

We now restrict the parameters so that dâ€” Â»ooand Hâ€”> 00 while

HW, (y ) = N/2d = p remains constant. As a consequence we have

(19) =, = B .

ns 3 ns'^s' - n Â»

-15-

thus the average scattering coefficient with one varia'ble held fixed

is independent of y ; i.e., no 'end effects* occur for an infinite

s

range. (The result (l9), which follows from elementary physical

considerations, can he deduced from (l8) with B -*oo, d -Â»-oo, if it is kept

in mind that P(y |y ,) as defined in (lo) is a function only of

S 8

|y -7 J.) In view of (l9) we may sum over s in (l?), replace y

hy y', and write

(20) W^pY: vVÂ°Â°Â«"''^'""' H^OcrOe^'^^'ay'. r- = [z^ ^(y-y.)^]' = ^ .

n -00

We reserve discussion of B for Section 3.1.

n

The integral over y' can he reduced to one evaluated exactly

12

hy Eeiche :

/CO I 5"

Ejc I 1^2 )

-00

1 + i v

' 1 + V

e-^^ dv

(21)

2 i

â– â– e

1/2 2

Kq - p

- ^n-

-1

INSTITUTE CF MATHEMATICAL SCIENCES

25 Wiverly Place, New York 3, N. "Â«.

NEW YORK UNIVERSITY

INSTITUTE OF MATHEMATICAL SCIENCES

DIVISION OF ELECTROMAGNETIC RESEARCH I

RESEARCH REPORT No. EM-58 !

MULTIPLE SCATTERING OF WAVES

BY PLANAR RANDOM DISTRIBUTIONS

OF PARALLEL CYLINDERS AND BOSSES

by

VIC TWERSKY

CONTRACT No. AF-19(122)-42

OCTOBER 1953

NEW YORK UNIVERSITY

Institute of Mathematical Sciences

Division of Electromagnetic Research

Research Report No. EM-58

MULTIPLE SCATTERING OF WAVES BY PLANAR RANDOM DISTRIBUTIONS

OF PARALLEL CYLINDERS AND BOSSES

by

Vic Twersky

Vi,c 'I'wersky^

Morris Kiine

Project Director

The research reported in this document has been made possible through

support and sponsorship extended by the Air Force Cambridge Research

Center, under Contract No. AF-19(122)-42. It is published for technical

information only, and does not necessarily represent recommendations

or conclusions of the sponsoring agency.

October, 1953

New York, 1953

Manufactured in the United States for New York University Press

by the University's Office of Publications and Printing

IA3LE or CONTEIITS

P8^

Abstract

1, Introduction ^

2, Preliminary Consideration and Statement of the

Problem ^

2.1 Scattering "by a Planar Configuration of

Parallel Cylinders 3

2.2 The Ensemljle of Configurations 9

2.3 Statement of the Problem ^3

3, The Average Wave Function ^^

3.1 Derivation of a Closed- form Approximation

for the Scattering Coefficients 17

3.2 Analysis of the Functions Involved 21

3.3 The Range of Validity of the Closed-fona

Representation 28

U, The Average Intensity 32

5. The Average Energy Flux ^1

6, Distribution of Bosses on a Perfectly Reflecting

Plane

Appendix

Abstract

56

The two-dimensional problem of multiple scattering of a plane

wave by a planar 'random' distribution of parallel cylinders is considered.

A formal solution,> 0, then the roots siaj' he approxi-

Tuated hy v^i^ (2WTi + c)/h. We find e = -i2inTp6, w^ ~ i2wrrp, and

consequently (l^) reduces to

00 iu2npu â–

(16) p(u)^p+^ ^^^iuznph â€¢ u>h^h.

tf= +1

We refer to (l6) as the 'more ordered' case. In the limit 6 â€” > 0, (l6)

reduces to the results for a diffraction grating of spacing "b.

2.3 Stfttftment of the Frohlem

We are now in a position to state explicitly the prohlem

considered in this paper. We are given the wave scattered hy a

single configuration of cylinders, i.e., xy = ^>/-Â» where vj/^ as

given "by (3) and (U) is expressed in terms of the known solution

for the single cylinder. We seek to average \|r and |\|;| over the

ensemhle specified "by the distrilmtion f-anction

*2^^8Â» ^s'^ ^ ^l^'^s^ ^^^s'^'s'^ ^Â°^ pairs where W^ and P are ^iven hy ^10)

and (ll). We also seek closed-form approximations for the averages

which may "be applied to the physical prohlem of scattered reflection

from a striated surface.

Sections 3 ^^^ ^ are concerned with the average values

of >p and |\|f| respectively. Section 5 deals with the average

energy fliix and certain energy theorems. Section 6 treats the

analogous proTslem for "bosses on a perfectly reflecting plane.

In the following sections we use either a "bar or angular

"brackets to indicate an average value.

^Ik-

3Â» The Average Wave S anction

The average value of the scattered wave defined "by (l),

(3) axid (^) niay he written as

8 3 J J

(17)

= E Z ^"^ / \ ^^8 ^^^ (-i'>Q^3sina)H^(fcr^ )e3p (inO^ ) < B^^>^ dy^

an "^ â€” d

where < B > indicates that B as in (^) is to he averaged over all

1^ s ns ^^

variahles except y , iÂ»e., over all configurations assumed hy cylinders

s' ^ 8. We refer to < > as an average with one variable held fixed.

(We use essentially the same notation and nomenclature as Foldy and

n

"Lblx â€¢ ) Similarly we have

^^, dy^. 1

where < B â€¢> t is the average scattering coefficient with two variables

OS SS '

held fixed.

We now restrict the parameters so that dâ€” Â»ooand Hâ€”> 00 while

HW, (y ) = N/2d = p remains constant. As a consequence we have

(19) =, = B .

ns 3 ns'^s' - n Â»

-15-

thus the average scattering coefficient with one varia'ble held fixed

is independent of y ; i.e., no 'end effects* occur for an infinite

s

range. (The result (l9), which follows from elementary physical

considerations, can he deduced from (l8) with B -*oo, d -Â»-oo, if it is kept

in mind that P(y |y ,) as defined in (lo) is a function only of

S 8

|y -7 J.) In view of (l9) we may sum over s in (l?), replace y

hy y', and write

(20) W^pY: vVÂ°Â°Â«"''^'""' H^OcrOe^'^^'ay'. r- = [z^ ^(y-y.)^]' = ^ .

n -00

We reserve discussion of B for Section 3.1.

n

The integral over y' can he reduced to one evaluated exactly

12

hy Eeiche :

/CO I 5"

Ejc I 1^2 )

-00

1 + i v

' 1 + V

e-^^ dv

(21)

2 i

â– â– e

1/2 2

Kq - p

- ^n-

-1

Online Library → Vic Twersky → Multiple scatterings of waves by planar random distributions of parallel cylinders and bosses → online text (page 1 of 4)